Journal of the
Korean Mathematical Society JKMS

Let $D$ be an integral domain with quotient field $K$, ${\bf X}$ be a nonempty set of indeterminates over $D$, $*$ be a star operation on $D$, $N_*=\{f \in D[{\bf X}]| c(f)^*=D\}$, $*_w$ be the star operation on $D$ defined by $I^{*_w} = ID[{\bf X}]_{N_*} \cap K$, and $[*]$ be the star operation on $D[{\bf X}]$ canonically associated to $*$ as in Theorem 2.1. Let $A^g$ (resp., $A^{*g}$, $A^{[*]g}$) be the global (resp., $*$-global, $[*]$-global) transform of a ring $A$. We show that $D$ is a $*_w$-Noetherian domain if and only if $D[{\bf X}]$ is a $[*]$-Noetherian domain. We prove that $D^{*g}[{\bf X}]_{N_*} = (D[{\bf X}]_{N_*})^g = (D[{\bf X}])^{[*]g}$; hence if $D$ is a $*_w$-Noetherian domain, then each ring between $D[{\bf X}]_{N_*}$ and $D^{*g}[{\bf X}]_{N_*}$ is a Noetherian domain. Let $\widetilde{D} = \cap\{D_P|P \in *_w$-Max$(D)$ and ht$P \geq 2\}$. We show that $D \subseteq \widetilde{D} \subseteq D^{*g}$ and study some properties of $\widetilde{D}$ and $D^{*g}$.

Keywords: star operation, $[*]$-operation on $D[{\bf X}]$, $*$-global transform, $*_w$-Noetherian domain, $D[{\bf X}]_{N_*}$, SM domain pair

MSC numbers: 13A15, 13B25, 13E99, 13G05